Step by step
The question
Question: find (f/g)(x) for f(x) = 3x + 2 and g(x) = 2x − 6.
(f/g)(x) means f of x divided by g of x.
We write it as a fraction: f on top, g at the bottom.
1 Put the rules in
Each rule goes in as one piece: (3x + 2)/(2x − 6).
2 Find what x can't be
We can never divide by 0.
So the bottom, g of x, can never be 0.
Careful!
The answer is not only the fraction.
x = 3 makes the bottom 0, so we must say it is not allowed.
Which x makes 2x − 6 zero?
We solve 2x − 6 = 0.
We want x by itself.
Two things are stuck to x: times 2 and − 6.
The − 6 is on the outside, so it comes off first.
Then the times 2.
3 Remove the − 6
To remove − 6, we add 6.
We do it on both sides, so they stay equal.
− 6 and + 6 make zero.
Cross them out.
0 + 6 = 6.
Now we have 2x = 6.
4 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
6 divided by 2 is 3.
So x = 3!
So (f/g)(x) = (3x + 2)/(2x − 6), with x ≠ 3.
5 Check
Let's check with a number: x = 1.
f of 1 is 5, g of 1 is −4.
Divided: −5/4.
That is our fraction at x = 1.
It works!
Let's look back.
First: put the rules in.
Then: find what x can't be.
Then: remove the − 6.
Then: remove the 2.
Last: check.